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Let $\mathfrak{u}$ denote the ultrafilter number, which is defined to be the minimum cardinality of a subset of $\mathcal{P}(\mathbb N)$ which is a base for a nonprincipal ultrafilter on $\mathbb{N}$. Clearly $\aleph_1\leq \frak{u}\leq 2^{\aleph_0}$, so it is only interesting to study $\frak{u}$ under the negation of CH. Kunen proved that it is consistent that CH fails and that $\frak{u}=\aleph_1$. Martin's axiom implies that $\frak{u}=2^{\aleph_0}$.

Is it consistent that $\aleph_1<\frak{u}<2^{\aleph_0}$? If so, can I please have a reference?

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The answer to your question is yes. In fact, one can force to make $\mathfrak u$ equal to any prescribed uncountable regular cardinal while making the cardinal of the continuum equal to any larger prescribed uncountable regular cardinal. This is proved in and old paper by Shelah and me:

Blass, Andreas(1-PAS); Shelah, Saharon(1-RTG) Ultrafilters with small generating sets. Israel J. Math. 65 (1989), no. 3, 259–271.

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  • $\begingroup$ Thanks Andreas. $\endgroup$
    – Isaac
    Commented Mar 12, 2019 at 23:48

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